# Theory StrongNorm

theory StrongNorm
imports LambdaType InductTermi
```(*  Title:      HOL/Proofs/Lambda/StrongNorm.thy
Author:     Stefan Berghofer
Copyright   2000 TU Muenchen
*)

section ‹Strong normalization for simply-typed lambda calculus›

theory StrongNorm imports LambdaType InductTermi begin

text ‹
Formalization by Stefan Berghofer. Partly based on a paper proof by
Felix Joachimski and Ralph Matthes @{cite "Matthes-Joachimski-AML"}.
›

subsection ‹Properties of ‹IT››

lemma lift_IT [intro!]: "IT t ⟹ IT (lift t i)"
apply (induct arbitrary: i set: IT)
apply (simp (no_asm))
apply (rule conjI)
apply
(rule impI,
rule IT.Var,
erule listsp.induct,
simp (no_asm),
simp (no_asm),
rule listsp.Cons,
blast,
assumption)+
apply auto
done

lemma lifts_IT: "listsp IT ts ⟹ listsp IT (map (λt. lift t 0) ts)"
by (induct ts) auto

lemma subst_Var_IT: "IT r ⟹ IT (r[Var i/j])"
apply (induct arbitrary: i j set: IT)
txt ‹Case @{term Var}:›
apply (simp (no_asm) add: subst_Var)
apply
((rule conjI impI)+,
rule IT.Var,
erule listsp.induct,
simp (no_asm),
simp (no_asm),
rule listsp.Cons,
fast,
assumption)+
txt ‹Case @{term Lambda}:›
apply atomize
apply simp
apply (rule IT.Lambda)
apply fast
txt ‹Case @{term Beta}:›
apply atomize
apply (simp (no_asm_use) add: subst_subst [symmetric])
apply (rule IT.Beta)
apply auto
done

lemma Var_IT: "IT (Var n)"
apply (subgoal_tac "IT (Var n °° [])")
apply simp
apply (rule IT.Var)
apply (rule listsp.Nil)
done

lemma app_Var_IT: "IT t ⟹ IT (t ° Var i)"
apply (induct set: IT)
apply (subst app_last)
apply (rule IT.Var)
apply simp
apply (rule listsp.Cons)
apply (rule Var_IT)
apply (rule listsp.Nil)
apply (rule IT.Beta [where ?ss = "[]", unfolded foldl_Nil [THEN eq_reflection]])
apply (erule subst_Var_IT)
apply (rule Var_IT)
apply (subst app_last)
apply (rule IT.Beta)
apply (subst app_last [symmetric])
apply assumption
apply assumption
done

subsection ‹Well-typed substitution preserves termination›

lemma subst_type_IT:
"⋀t e T u i. IT t ⟹ e⟨i:U⟩ ⊢ t : T ⟹
IT u ⟹ e ⊢ u : U ⟹ IT (t[u/i])"
(is "PROP ?P U" is "⋀t e T u i. _ ⟹ PROP ?Q t e T u i U")
proof (induct U)
fix T t
assume MI1: "⋀T1 T2. T = T1 ⇒ T2 ⟹ PROP ?P T1"
assume MI2: "⋀T1 T2. T = T1 ⇒ T2 ⟹ PROP ?P T2"
assume "IT t"
thus "⋀e T' u i. PROP ?Q t e T' u i T"
proof induct
fix e T' u i
assume uIT: "IT u"
assume uT: "e ⊢ u : T"
{
case (Var rs n e1 T'1 u1 i1)
assume nT: "e⟨i:T⟩ ⊢ Var n °° rs : T'"
let ?ty = "λt. ∃T'. e⟨i:T⟩ ⊢ t : T'"
let ?R = "λt. ∀e T' u i.
e⟨i:T⟩ ⊢ t : T' ⟶ IT u ⟶ e ⊢ u : T ⟶ IT (t[u/i])"
show "IT ((Var n °° rs)[u/i])"
proof (cases "n = i")
case True
show ?thesis
proof (cases rs)
case Nil
with uIT True show ?thesis by simp
next
case (Cons a as)
with nT have "e⟨i:T⟩ ⊢ Var n ° a °° as : T'" by simp
then obtain Ts
where headT: "e⟨i:T⟩ ⊢ Var n ° a : Ts ⇛ T'"
and argsT: "e⟨i:T⟩ ⊩ as : Ts"
by (rule list_app_typeE)
from headT obtain T''
where varT: "e⟨i:T⟩ ⊢ Var n : T'' ⇒ Ts ⇛ T'"
and argT: "e⟨i:T⟩ ⊢ a : T''"
by cases simp_all
from varT True have T: "T = T'' ⇒ Ts ⇛ T'"
by cases auto
with uT have uT': "e ⊢ u : T'' ⇒ Ts ⇛ T'" by simp
from T have "IT ((Var 0 °° map (λt. lift t 0)
(map (λt. t[u/i]) as))[(u ° a[u/i])/0])"
proof (rule MI2)
from T have "IT ((lift u 0 ° Var 0)[a[u/i]/0])"
proof (rule MI1)
have "IT (lift u 0)" by (rule lift_IT [OF uIT])
thus "IT (lift u 0 ° Var 0)" by (rule app_Var_IT)
show "e⟨0:T''⟩ ⊢ lift u 0 ° Var 0 : Ts ⇛ T'"
proof (rule typing.App)
show "e⟨0:T''⟩ ⊢ lift u 0 : T'' ⇒ Ts ⇛ T'"
by (rule lift_type) (rule uT')
show "e⟨0:T''⟩ ⊢ Var 0 : T''"
by (rule typing.Var) simp
qed
from Var have "?R a" by cases (simp_all add: Cons)
with argT uIT uT show "IT (a[u/i])" by simp
from argT uT show "e ⊢ a[u/i] : T''"
by (rule subst_lemma) simp
qed
thus "IT (u ° a[u/i])" by simp
from Var have "listsp ?R as"
by cases (simp_all add: Cons)
moreover from argsT have "listsp ?ty as"
by (rule lists_typings)
ultimately have "listsp (λt. ?R t ∧ ?ty t) as"
by simp
hence "listsp IT (map (λt. lift t 0) (map (λt. t[u/i]) as))"
(is "listsp IT (?ls as)")
proof induct
case Nil
show ?case by fastforce
next
case (Cons b bs)
hence I: "?R b" by simp
from Cons obtain U where "e⟨i:T⟩ ⊢ b : U" by fast
with uT uIT I have "IT (b[u/i])" by simp
hence "IT (lift (b[u/i]) 0)" by (rule lift_IT)
hence "listsp IT (lift (b[u/i]) 0 # ?ls bs)"
by (rule listsp.Cons) (rule Cons)
thus ?case by simp
qed
thus "IT (Var 0 °° ?ls as)" by (rule IT.Var)
have "e⟨0:Ts ⇛ T'⟩ ⊢ Var 0 : Ts ⇛ T'"
by (rule typing.Var) simp
moreover from uT argsT have "e ⊩ map (λt. t[u/i]) as : Ts"
by (rule substs_lemma)
hence "e⟨0:Ts ⇛ T'⟩ ⊩ ?ls as : Ts"
by (rule lift_types)
ultimately show "e⟨0:Ts ⇛ T'⟩ ⊢ Var 0 °° ?ls as : T'"
by (rule list_app_typeI)
from argT uT have "e ⊢ a[u/i] : T''"
by (rule subst_lemma) (rule refl)
with uT' show "e ⊢ u ° a[u/i] : Ts ⇛ T'"
by (rule typing.App)
qed
with Cons True show ?thesis
by (simp add: comp_def)
qed
next
case False
from Var have "listsp ?R rs" by simp
moreover from nT obtain Ts where "e⟨i:T⟩ ⊩ rs : Ts"
by (rule list_app_typeE)
hence "listsp ?ty rs" by (rule lists_typings)
ultimately have "listsp (λt. ?R t ∧ ?ty t) rs"
by simp
hence "listsp IT (map (λx. x[u/i]) rs)"
proof induct
case Nil
show ?case by fastforce
next
case (Cons a as)
hence I: "?R a" by simp
from Cons obtain U where "e⟨i:T⟩ ⊢ a : U" by fast
with uT uIT I have "IT (a[u/i])" by simp
hence "listsp IT (a[u/i] # map (λt. t[u/i]) as)"
by (rule listsp.Cons) (rule Cons)
thus ?case by simp
qed
with False show ?thesis by (auto simp add: subst_Var)
qed
next
case (Lambda r e1 T'1 u1 i1)
assume "e⟨i:T⟩ ⊢ Abs r : T'"
and "⋀e T' u i. PROP ?Q r e T' u i T"
with uIT uT show "IT (Abs r[u/i])"
by fastforce
next
case (Beta r a as e1 T'1 u1 i1)
assume T: "e⟨i:T⟩ ⊢ Abs r ° a °° as : T'"
assume SI1: "⋀e T' u i. PROP ?Q (r[a/0] °° as) e T' u i T"
assume SI2: "⋀e T' u i. PROP ?Q a e T' u i T"
have "IT (Abs (r[lift u 0/Suc i]) ° a[u/i] °° map (λt. t[u/i]) as)"
proof (rule IT.Beta)
have "Abs r ° a °° as →⇩β r[a/0] °° as"
by (rule apps_preserves_beta) (rule beta.beta)
with T have "e⟨i:T⟩ ⊢ r[a/0] °° as : T'"
by (rule subject_reduction)
hence "IT ((r[a/0] °° as)[u/i])"
using uIT uT by (rule SI1)
thus "IT (r[lift u 0/Suc i][a[u/i]/0] °° map (λt. t[u/i]) as)"
by (simp del: subst_map add: subst_subst subst_map [symmetric])
from T obtain U where "e⟨i:T⟩ ⊢ Abs r ° a : U"
by (rule list_app_typeE) fast
then obtain T'' where "e⟨i:T⟩ ⊢ a : T''" by cases simp_all
thus "IT (a[u/i])" using uIT uT by (rule SI2)
qed
thus "IT ((Abs r ° a °° as)[u/i])" by simp
}
qed
qed

subsection ‹Well-typed terms are strongly normalizing›

lemma type_implies_IT:
assumes "e ⊢ t : T"
shows "IT t"
using assms
proof induct
case Var
show ?case by (rule Var_IT)
next
case Abs
show ?case by (rule IT.Lambda) (rule Abs)
next
case (App e s T U t)
have "IT ((Var 0 ° lift t 0)[s/0])"
proof (rule subst_type_IT)
have "IT (lift t 0)" using ‹IT t› by (rule lift_IT)
hence "listsp IT [lift t 0]" by (rule listsp.Cons) (rule listsp.Nil)
hence "IT (Var 0 °° [lift t 0])" by (rule IT.Var)
also have "Var 0 °° [lift t 0] = Var 0 ° lift t 0" by simp
finally show "IT …" .
have "e⟨0:T ⇒ U⟩ ⊢ Var 0 : T ⇒ U"
by (rule typing.Var) simp
moreover have "e⟨0:T ⇒ U⟩ ⊢ lift t 0 : T"
by (rule lift_type) (rule App.hyps)
ultimately show "e⟨0:T ⇒ U⟩ ⊢ Var 0 ° lift t 0 : U"
by (rule typing.App)
show "IT s" by fact
show "e ⊢ s : T ⇒ U" by fact
qed
thus ?case by simp
qed

theorem type_implies_termi: "e ⊢ t : T ⟹ termip beta t"
proof -
assume "e ⊢ t : T"
hence "IT t" by (rule type_implies_IT)
thus ?thesis by (rule IT_implies_termi)
qed

end
```